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Two-Count Filter Method for Measurement 

of Thoron Progeny in Air

ORAUT-RPRT-0084 Rev. 00

Effective Date: 03/27/2017

Supersedes: None

Subject Expert(s): Thomas R. LaBone

Document Owner 

Approval: Signature on File

Jennifer L. Hoff, Document Owner

Approval Date: 03/17/2017

Concurrence: Vickie S. Short Signature on File for

James P. Griffin, Deputy Project Director

Concurrence Date: 03/17/2017

Concurrence: Vickie S. Short Signature on File for

Kate Kimpan, Project Director

Concurrence Date: 03/17/2017

Approval: Signature on File

James W. Neton, Associate Director for Science

Approval Date: 03/27/2017

FOR DOCUMENTS MARKED AS A TOTAL REWRITE, REVISION, OR PAGE CHANGE, REPLACE THE PRIOR 

REVISION AND DISCARD / DESTROY ALL COPIES OF THE PRIOR REVISION.

New Total Rewrite Revision Page Change

DOE Review Release 03/30/2017
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 2 of 14

PUBLICATION RECORD

EFFECTIVE 

DATE

REVISION

NUMBER DESCRIPTION

03/27/2017 00 New document initiated to describe a two-count filter method for 

measurement of thoron progeny in air. Incorporates formal internal 

and NIOSH review comments. Training required: As determined by 

the Objective Manager. Initiated by Thomas R. LaBone.
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 3 of 14

TABLE OF CONTENTS

SECTION TITLE PAGE

Acronyms and Abbreviations..................................................................................................................4

1.0 Introduction .................................................................................................................................5

2.0 The Forward Problem .................................................................................................................6

3.0 The Reverse Problem.................................................................................................................9

References ...........................................................................................................................................11

ATTACHMENT A EXACT SOLUTION TO THE FORWARD PROBLEM.......................................12

LIST OF FIGURES

FIGURE TITLE PAGE

1-1 232Th decay chain........................................................................................................................5

2-1 Deposition of 212Pb on the air filter..............................................................................................7
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 4 of 14

 

ACRONYMS AND ABBREVIATIONS 

GKE general kinetics equation

hr hour

L liter

min minute

NIOSH National Institute for Occupational Safety and Health

ORAU Oak Ridge Associated Universities

pCi picocurie

SRDB Ref ID Site Research Database Reference Identification (number)
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 5 of 14

 

1.0 INTRODUCTION

Thoron (220Rn) is an inert radioactive gas that is a member of the thorium (232Th) decay chain as 

shown in Figure 1-1. 

Figure 1-1. 232Th decay chain (Wikipedia 2017). 

As discussed in DCAS-TIB-011, Lung Dose Conversion Factors for Thoron WLM (NIOSH 2013), 212Bi 

and 212Pb are the primary radionuclides of interest when calculating the internal dose from inhalation 

of thoron and its progeny. The two-count filter method can be used to estimate the concentration of 
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 6 of 14

 

212Pb in ambient air. The method consists of pulling air through a filter for a prescribed length of time 

and measuring the total alpha activity on the filter at 6 hours after the pump stops and again at 

24 hours after the pump stops. Given these two alpha activities, the concentration of 212Pb in the air 

can be calculated when long-lived alpha emitters and radon (222Rn) progeny1 are also present in the 

air. 

The goal here is to derive an equation for the two-count filter method. The approach is two-fold: 

1. Start with known concentrations of 212Pb, 212Bi, and a long-lived alpha emitter in the air and 

from this calculate the total alpha activity expected to be on the filter paper when it is counted 

at 6 and 24 hours after the air sampler pump is turned off. This is referred to as the “forward 

problem,” and its end product is a benchmark solution.

2. Work back from the total alpha activity on the filter at 6 and 24 hours and derive an equation 

that gives the concentration of 212Pb in the air. This is referred to as the “reverse problem.” 

If the equation derived in the reverse problem gives the same 212Pb air concentration as used in the 

forward problem, the equation is deemed to be correct. 

In summary, the primary purpose of this report is to provide the technical basis for Equation 3-7, 

which is used to estimate the concentration of 212Pb in air given alpha counts of the air filter at 6 and 

24 hours after the filter pump is stopped.

2.0 THE FORWARD PROBLEM

The source term is assumed to be a constant stream of 212Pb and 212Bi into the air sampler, and the 

concentration of each radionuclide is assumed to be 1 pCi/L. The goal is to calculate total alpha 

activity that will be on the filter at 6 and 24 hours after the sampler pump is turned off. Terms used in 

the calculations are:

F = flow rate of sampler (15 L/min = 900 L/hr)

T = length of time pump is run (24 hr) 

λpb = decay constant for 212Pb (0.06513/hr) 

λbi = decay constant for 212Bi (0.6869/hr) 

Cpb = 212Pb concentration in the air going into the sampler (1 pCi/L)

t = variable time (hr) 

dt = infinitesimally short time increment (hr) 

As shown in Figure 2-1, at time t a very small amount of 212Pb (the product of Cpb, F, and dt) has 

deposited on the filter. Between times t and T when the pump is turned off, this small amount of 212Pb 

will have decayed to 212Bi. Equation 2-1 sums the small depositions that take place at all times 

 

1 The radon progeny on the filter are assumed to have completely decayed away by the time the filter is counted for the 

first time.
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between the time the pump is started and the time it is turned off to give the 212Pb activity Apb0 on the 

filter at time T: 

(2-1)

( )

( )

=

− −

=

=

−

=

−

  =      

=

= −

∫

∫

pb

pb pb

pb

pb0

0

0

pb

1

t T

λ Tt

pb

t

t T

λT λt

pb

t

pb λ T

A C Fdt e

C Fe e dt

C F

e

λ

Figure 2-1. Deposition of 212Pb (Apb0) on the air filter. 

Substituting the values of the parameters into Equation 2-12 gives the activity of 212Pb on the filter 

when the pump stops:

(2-2) 4 Apb0 = × 1.092 10 pCi

 

2 In the numerical examples given in this paper the conversion of units is implicit, i.e., unit conversion constants have 

been omitted for clarity. 
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 8 of 14

 

The 212Pb activities on the filter at 6 and 24 hours after the pump stops are: 

( ) (2-3) 

( ) ( )

−

− −

− −+

=

= −

= −

pb

pb pb

pb pb

6

pb6 pb0

6

pb

6 6

pb

1

λ

pb λT λ

pb λ λT

A Ae

C F

e e

λ

C F

e e

λ

( ) (2-4) 

( ) ( )

−

− −

− −+

=

= −

= −

pb

pb pb

pb pb

24

pb24 pb0

24

pb

24 24

pb

1

λ

pb λT λ

pb λ λT

A Ae

C F

e e

λ

C F

e e

λ

At the times of the 6-hour and 24-hour counts, the 212Bi activities are assumed to be in transient 

equilibrium with the 212Pb activities, i.e., the unsupported 212Bi on the filter is insignificant in 

comparison with the activity of supported 212Bi and can be ignored in this application. Another way to 

state this is that the two-filter method with counts at 6-hours and 24-hours after the pump stops is 

insensitive to the concentration of 212Bi in the air and cannot provide information on the concentration 

of that thoron progeny.

Therefore, the ratios of 212Bi to 212Pb activity on the filter at these times is: 

(2-5) bi

bi pb

1 105 λr .

λ λ = = −

The 212Bi activities on the filter at 6 and 24 hours are therefore: 

(2-6) ( ) ( ) − −+    = =    − −   

bi pb 6 6 pb

bi6 pb6

bi pb pb

λ C Fpb λ λT A rA e e

λλ λ

(2-7) ( ) ( ) − −+    = =    − −   

bi pb 24 pb 24

bi24 pb24

bi pb pb

λ C Fpb λ λT A rA e e

λλ λ

Because the alpha particles are emitted by 212Bi (not 212Pb), the total alpha activities on the filter at 

6 and 24 hours are: 

(2-8) 6 bi6 α AA A = +

(2-9) AA A 24 bi24 = + α
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 9 of 14

where Aα is the activity from a long-lived alpha emitter on the filter, which is not a parameter of interest 

in this application3. 

For the benchmark problem, it is assumed that: 

( )( )( ) (2-10) α

60 min 0.05 pCi/L 15 L/min 24 hr 1,080 pCi hr

  = =     A

Therefore, the total alpha activities on the filter at 6 and 24 hours after the pump is stopped are: 

(2-11) ( ) ( ) − −+   

=    − += −   

bi pb 6 6 pb

6 α

bi pb pb

9 245 pCi λ C Fpb λ λT A e e A, λλ λ

( ) ( ) − −+   

=    − += −   

bi pb 24 pb 24

24 α

bi pb pb

3 608 pCi λ C Fpb λ λT A e e A, λλ λ (2-12) 

Ultimately, an equation will be derived that takes these alpha activities and returns the concentration 

of 212Pb in the air. If a concentration of 1 pCi/L is returned, it can be taken as evidence that the 

equation faithfully implements the model and that the math has been done correctly. 

3.0 THE REVERSE PROBLEM

In the reverse problem, the total alpha activity on the filter at 6 and 24 hours after the pump is turned 

off is known, and the goal is to calculate the concentration of 212Pb in the air. From Equations 2-8 and 

2-9, it can be concluded that:

(3-1) 24 bi24 6 bi6 A A AA − =−

Then

(3-2) A rA A rA 24 pb24 6 pb6 − =−

Note that the activity of the long-lived alpha emitter drops out of the equations as desired. This 

relationship can be rearranged to produce: 

(3-3) ( )

( ) pb pb

24 6 pb24 pb6

24 6

pb0 pb0

λ λ

A A rA A

rA e A e − −

−= −

= −

Then

(3-4) ( ) pb pb

24 6

pb0 λ λ 24 6

A A A

re e − −

− = −

3 At some facilities there was interest in the long-lived alpha emitter, and the two-count method was used in the opposite 

way to strip out thoron background.
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From Equation 2-1 it is known that: 

( ) (3-5) − = − pb

pb0

pb

1

C Fpb λ T A e

λ

Equating the right-hand sides of Equations 2-1 and 3-4 gives: 

(3-6) ( ) ( ) −

− −

− = −

−

pb

pb pb

24 6

24 6

pb

1 pb λ T

λ λ

A A C F

e

re e λ

which can be solved for Cpb, the 212Pb concentration in the sampled air:

(3-7) ( ) − − −

    −   − =           −   −  

pb pb pb

bi pb 24 6 pb

24 6

bi 1 pb λ λ λ T

λ λ A A λ

C

λ e e F e

When the numbers are put into Equation 3-7 the result is: 

= 1.0 pCi/L (3-8) Cpb

which confirms that the calculation was done correctly.

In the derivation of Equation 3-7, it is assumed that the 212Bi on the filter was in transient equilibrium 

with the 212Pb when the 6- and 24-hour counts were performed. As discussed before, this means that 

no unsupported 212Bi was present on the filter when the counts were made. Both assumptions are 

eminently reasonable considering the half-lives of 212Pb and 212Bi and the times involved.4 These 

assumptions should be checked if the filter is counted less than 6 hours after the pump is stopped.

4 The validity of both assumptions was also confirmed with more detailed calculations that are given in Attachment A.
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REFERENCES 

NIOSH (National Institute for Occupational Safety and Health), 2013, Lung Dose Conversion Factor 

for Thoron WLM, DCAS-TIB-011, Rev. 4, Division of Compensation Analysis and Support,

Cincinnati, Ohio, May 1. [SRDB Ref ID: 126391]

Skrable, K. W., C. French, G. Chabot, and A. Major, 1974, “General Equation for the Kinetics of 

Linear First Order Phenomena and Suggested Applications,” Health Physics, volume 27, pp. 

155–157. [SRDB Ref ID: 165844] 

Wikipedia, 2017, “Thorium,” Wikimedia Foundation, San Francisco, California, February 17. [SRDB 

Ref ID: 165836] 
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 12 of 14

 

ATTACHMENT A 

EXACT SOLUTION TO THE FORWARD PROBLEM

The calculations given above use the assumption that 212Bi and 212Pb are in transient equilibrium at 

the times of interest (i.e., the 6- and 24-hour counts). This greatly simplifies the calculations because 

it is not necessary to explicitly model the decay of the 212Bi. Here, Skrable’s general kinetics equation

(GKE) (Skrable et al. 1974) is used to derive the 6- and 24-hour filter activities without assuming 

transient equilibrium.

The GKE shown below is generally applicable to first-order processes such as radioactive decay and 

the biokinetics of assimilated material, so the notation is somewhat generic (and perhaps imposing). 

Here, the GKE is applied to members (species) of a radioactive decay chain, so the notation and the 

equations will simplify significantly.

(A-1) 

( )

( )

( )

− − −

+

= = =

= =

≠ ≠

       

  − =   +

  − −

   

∑ ∑ ∏

∏ ∏

1 0

, 1

1

1 j j

k t k t n n n i i

n j j n n

i j i j i

pj j pj

p i p i

p j p j

N e P e

N k

kk k kk

where

Pi = atom production rate for the ith species. For the problem, this is the rate at which 

atoms of 212Pb and 212Bi are deposited on the filter.

Ni

0 = the number of atoms of the ith species present at time 0.

Ni = the number of atoms of the ith species present at time t. 

kj,j+1 = transfer rate constant from the jth species to the j+1 species/compartment. Here these 

are the radioactive decay constants.

kj = total removal rate constant from the jth species. Here these are the decay constants 

and are the same as the transfer rate constants (i.e., kj = kj,j+1). 

Note that the GKE deals with atoms, and activity is calculated once the final solution in terms of atoms 

is obtained. 

The calculation is broken down into two parts: the first is over the time from when the pump is turned 

on to the time when it is stopped, and the second is at times after the pump has stopped.

When the pump is started there are no atoms of 212Pb or 212Bi on the filter, so Ni

0 = 0 for all members 

of the decay chain. The GKE in this case reduces to: 

(A-2) ( )

( )

1

, 1

1

1 j k t n n n i

n j j n

i j i j i

j pj

p i

p j

P e

N k

k kk

− −

+

= = =

=

≠

       

    − =         −  

   

∑ ∑ ∏

∏
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 13 of 14

ATTACHMENT A 

EXACT SOLUTION TO THE FORWARD PROBLEM

For 212Pb, set k1 = λpb, k2 = λbi, t = T (i.e., the pump has just been shut off): 

= = = × (A-3) 6

1 pb

pb

1 841 10 / hr

C Fpb PP .

λ

P1 is the number of atoms per hour deposited on filter, and N1 = Npb0. After working through all the 

summations and pi products the result is: 

( ) (A-4) pb pb 7

pb0

pb

1 2 234 10

P λ T N e.

λ

− =− = ×

Therefore: 

(A-5) 4 Apb0 pb pb0 = = × λN . 1 092 10 pCi

which is the same result for the 212Pb activity on the filter given previously in Section 2.0. Similarly, for 

212Bi: 

= = = × (A-6) 5

2 bi

bi

1 745 10 /hr C Fbi PP .

λ

and N2 = Nbi0. In this case the GKE gives: 

( ) (A-7) ( )

( )

( )

( )

− −

−

  − − =− +   + = × − −  

pb bi

bi pb pb bi 6

bi0

bi pb bi pb bi pb bi

1 1

1 2 314 10

λ T λ T

λ T

pb

P P e P e

N e λ .

λ λ λ λ λλ λ

Abi0 was not calculated before because it was not needed, but for completeness it is shown here: 

(A-8) 4 Abi0 bi bi0 = = × λN . 1 193 10 pCi

After the pump is stopped the production rates for 212Pb and 212Bi are equal to zero. The GKE with 

P = 0 for all members of the decay chain is: 

(A-9)

( )

1 0

, 1

1

j k t n n n

i

n j j n

i j i j i

p j

p i

p j

N e N k

k k

− −

+

= = =

=

≠

       

    =         −        

∑ ∑ ∏

∏

Now all times are relative to the time the pump was stopped. That is, N0 for the first species = Npb0

and N0 for the second species = Nbi0. The GKE for 212Pb reduces to: 

(A-10) pb

pb pb0

λ t N Ne t

− =
Document No. ORAUT-RPRT-0084 Revision No. 00 Effective Date: 03/27/2017 Page 14 of 14

ATTACHMENT A 

EXACT SOLUTION TO THE FORWARD PROBLEM

where Npbt is the number of atoms of 212Pb on the filter at time t after the pump was shut off. For 212Bi: 

(A-11) ( ) ( )

0 0

0

pb bi

bi

t t

t pb pb

bit bi pb

bi pb pb bi

Ne Ne

N Ne

λ λ

λ λ

λλ λ λ

− −

−  

=+ +   − −  

where Nbit is the number of atoms of 212Bi on the filter at time = t after the pump was shut off. 

For t = 6 hours:

(A-12) 7

pb6 N = × 1.512 10

(A-13) 6 Nbi6 = × 1.583 10

For t = 24 hours:

(A-14) 6

pb24 N = × 4.681 10

(A-15) 5 Nbi24 = × 4.903 10

The corresponding activities are:

(A-16) Apb6 pb pb6 = = λN , 7 391 pCi

(A-17) Abi6 bi bi6 = = λN , 8 163 pCi

(A-18) Apb24 pb pb24 = = λN , 2 289 pCi

(A-19) Abi24 bi bi24 = = λN , 2 528 pCi

Referring back to Equations 2-8 and 2-9 and using the values for Abi6 and Abi24 calculated above using 

the GKE the results are: 

(A-20) AA A 6 bi6 = += + = α 8,163 pCi 1,080 pCi 9,243 pCi

(A-21) AA A 24 bi24 = += + = α 2,528 pCi 1,080 pCi 3,608 pCi

which agree with the previously calculated values under the assumption of transient equilibrium.